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Properties

Label 2-912-304.107-c1-0-73
Degree $2$
Conductor $912$
Sign $-0.192 + 0.981i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.520 + 1.31i)2-s + (0.965 + 0.258i)3-s + (−1.45 + 1.36i)4-s + (0.825 − 3.08i)5-s + (0.162 + 1.40i)6-s − 4.77·7-s + (−2.55 − 1.20i)8-s + (0.866 + 0.499i)9-s + (4.48 − 0.517i)10-s + (0.955 − 0.955i)11-s + (−1.76 + 0.944i)12-s + (−6.88 + 1.84i)13-s + (−2.48 − 6.27i)14-s + (1.59 − 2.76i)15-s + (0.253 − 3.99i)16-s + (0.521 + 0.903i)17-s + ⋯
L(s)  = 1  + (0.367 + 0.929i)2-s + (0.557 + 0.149i)3-s + (−0.729 + 0.684i)4-s + (0.369 − 1.37i)5-s + (0.0662 + 0.573i)6-s − 1.80·7-s + (−0.904 − 0.426i)8-s + (0.288 + 0.166i)9-s + (1.41 − 0.163i)10-s + (0.288 − 0.288i)11-s + (−0.508 + 0.272i)12-s + (−1.90 + 0.511i)13-s + (−0.663 − 1.67i)14-s + (0.411 − 0.713i)15-s + (0.0633 − 0.997i)16-s + (0.126 + 0.219i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.192 + 0.981i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (715, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.192 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.289086 - 0.351281i\)
\(L(\frac12)\) \(\approx\) \(0.289086 - 0.351281i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.520 - 1.31i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
19 \( 1 + (3.97 + 1.79i)T \)
good5 \( 1 + (-0.825 + 3.08i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + 4.77T + 7T^{2} \)
11 \( 1 + (-0.955 + 0.955i)T - 11iT^{2} \)
13 \( 1 + (6.88 - 1.84i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.521 - 0.903i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.680 + 1.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.35 + 1.43i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + 7.22T + 31T^{2} \)
37 \( 1 + (-2.28 + 2.28i)T - 37iT^{2} \)
41 \( 1 + (1.92 + 3.32i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.06 + 0.553i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (11.6 + 6.71i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.05 + 3.95i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.14 - 4.27i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.22 - 8.28i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.128 - 0.480i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.16 - 3.56i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.49 + 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.32 - 9.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.0 - 11.0i)T + 83iT^{2} \)
89 \( 1 + (-6.47 + 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.1 - 7.04i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.498256344661276021877682103240, −9.052210341487657334212048953656, −8.305413359462520524605306970662, −7.11921173925574992280990331454, −6.51434334563579701109326780352, −5.43431720443390065555235933580, −4.61779698514428278999829624567, −3.69977224407301807546563647528, −2.49244030831478819133193183636, −0.16158428044532167056729631139, 2.17904160970952246259491542960, 2.96950368670177184723190115035, 3.46417954999832834618580288860, 4.85029808113999955608359809838, 6.21754726209945476139019920227, 6.70581626480023148568722570844, 7.68964465752145017710117419356, 9.143276663590960969633961875641, 9.883538389237443335825321444482, 10.04956964935649241055391112911

Graph of the $Z$-function along the critical line